Monday, 18 April 2016

10 - The Toric Code (part 5)

This is part of a project to get people involved with quantum error correction. See here for more info.I'm
going to jump straight in with this post. If you want an intro, see
part 4.

Protecting
against measurements

We
have information stored in a toric code, which is a grid of noisy
qubits. If they get hit by bit flips, its mostly fine. The only way
for bad things to happen is if a load of bit flips happen all at the
same time, and do so on a loop that goes around the doughnut. That's
unlikely, so we're mostly okay.

Bad
things also happen if the qubits get measured on a loop through the
doughnut. Currently we have no way of detecting or correcting these,
so they will build up over time and soon mess up our logical qubit.
So how can we manage them?

Currently
we have a rule for each white square: each must have an even number
of 1s around it. By looking at which of these rules are broken, we
can work out how to get rid of the bit flip errors. But let's forget
about these rules for a while. Instead we will come up with some new
rules that can be used to detect the measurements. Once we've worked
those out, we can try the two sets of rules play well together.

Rules
for blue plaquettes

Let's
now think about the code using the + and – states from last time.
With these we can declare a rule for each blue squares: when looked
at in terms of + and -, each must have an even number of -'s (and
therefore an even number of +'s).

There
are many possible patterns of +'s and -'s that obey these rules.
Here's a few.

These
should look pretty familiar. They are the same ones as before, but
with 0 replaced with +, 1 to –, white squares with blue and
everything turned 90 degrees.

Now
we go off and get a cup of tea, and a Gremlin sees an opportunity for
some mischief. It measures one of the qubits to see if it is 0 or 1.
Every qubit here is + or -, both of which are a superposition of 0
and 1. So the measured qubit will randomly choose one or the other.
Let's say, for sake of argument, it chooses 0.

When
we get back from our tea break, we check whether the blue squares are
following the rules. Most are, but what about the two near the
measured qubit?

When
we measure those squares, we are asking them to look at how many of
the qubits are in state +, and how many are in state -, and tell us
whether the number of -'s is odd or even. For most of the qubits,
which are simply in state + or -, this is an easy job. But the
measured one is in state 0, a superposition of + and -. This has to
decide whether to be + or – before the question can be answered.

If
it decides to be +, which is what this qubit was originally, the
rules are obeyed on these squares. The qubit has also been forced
into a state that is nothing like the 0 state that the Gremlin
measured, so that information is now out of date and the Gremlin
knows nothing. All is well.

If
it decides to be the opposite of what it was originally, the effect
is similar to a bit flip. Except this kind of flip turns + to – and
– to +. We call these phase flips. The phase flip causes the
rules to be broken on the two blue squares. But, just like with bit
flips and white squares, this allows us to detect and correct the
errors. Also, the qubit is again forced into a state that has nothing
to do with the one that the Gremlin measured. So its information is
again out of date, and all is well.

With
these rules, protecting the code from measurements is exactly the
same as protecting it from bit flips. So a few measurements here and
there don't cause a big problem. We can just detect them, remove
their effects and update the state to make sure their results can't
tell anyone anything about our logical qubit. The only way for an
unwanted measurement to happen is if a load of measurements happen
all at the same time, and do so on a loop that goes through the
doughnut. That's unlikely, so we're mostly okay.

Combining
the rules

We
now have a set of rules for dealing with bit flips, and a set for
dealing with phase flips. But we really need them to work together,
so we can deal with both at the same time.

Getting
things to work together in quantum mechanics is not always possible.
Suppose we have a single qubit, and set two rules:

i)
When we measure whether it is 0 or 1, the answer must always come out
0.

ii)
When we measure whether it is + or -, the answer must always come out +.

The
only way to make a qubit always give us the result 0 for a 0/1
measurement, as rule (i) requires, is to have it in state 0. But this
is a superposition of + and -. So it gives the result – half the
time when we do a +/- measurement, breaking rule (ii). Similarly, the
+ state is the only one that always obeys rule (ii), but it breaks
(i) half the time.

If
the same was true for the rules for the squares, we would get loads
of broken rules all the time. It would be impossible to get enough
information to detect and correct the errors, because of all the
random nonsense that comes from the rules fighting each other.

But
there is a big difference between rules (i) and (ii), and the rules
we use in the toric code. There is not just one state that satisfies
the rules for the white squares, but there are many patterns of 0s
and 1s that do it. And any superposition of these will obey them too.
There are also many patterns of +'s and -'s that obey the blue square
rules, as do their superpositions. With more possibilities, there is
hope.

Any
state can be thought of either in terms of 0s and 1s or in terms of
+'s and -'s. Just like a single qubit in state 0 can be thought of as
just a 0, or as a superposition of + and -. So it could be that some
superposition of white rule abiding 0s and 1s turns out to be the
same state as some superposition of blue rule abiding +'s and -'s. If
so, states like this would obey both the rules for white squares and
blue squares at once.

Fortunately,
it turns out that there are states like this. They are not just nice
simple patterns of +'s and -'s as in the figure above, or nice simple
patterns of 0s and 1s as in the figures for previous posts. We need a
big superposition to make sure that the rules can work well together.
In fact, for a logical 0 we need a superposition of every possible
set of 0s and 1s that could be used for a logical 0. Making such a
complicated set of correlated qubits is going to be quite hard, and
is the main reason that you can't buy a toric code from the shops at
the moment. But experimentalists are working hard towards this goal..

With
both sets of rules working well together, we just need to keeping
checking on both types of square. When we see white
rules
broken, we know that a
bit flip
has happened, and can correct it. When blue rules are broken, we
detect and correct
phase flips.
As long as unlikely things don't happen, or logical qubit will keep
nice and safe.

Phase
flips and +/- measurements

So
far we've assumed that the only kind of bad things that happen are
bit flips and unwanted measurements of 0 and 1. But what about phase
flips, or unwanted measurements of + and -? Our code naturally sorts
those out too. What about other crazy weird quantum stuff? It turns
out that anything that could ever happen can be thought of as some
combination of bit and phase flips. So whatever happens, once we
measure what rules are being obeyed, the noise gets forced to decide
what kind of bit and phase flips it wants to be. The we detect and
correct those, and all is well. The toric code (as well as other
quantum error correcting codes) protects us from everything.

Summary

Well,
perhaps it doesn't protect is from everything: big and horrible
errors that hit us all at once can still get under the radar. But as
long as they are rare, our logical qubit will last a long time. And
if they are not rare enough, we just make our code bigger so they
need to be ever bigger and more horrible before they can mess us up.

So
that's the toric code in 5 parts. Next we will move on to its flat
brother, the planar code. But first, in this post we take a look at some of the
maths behind all this. I try to keep it with little maths as
possible, but it did turn out to be a bit long. There is a bit of teleportation in there, though.